8z+22=3(3z+11)z

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Solution for 8z+22=3(3z+11)z equation:


Simplifying
8z + 22 = 3(3z + 11) * z

Reorder the terms:
22 + 8z = 3(3z + 11) * z

Reorder the terms:
22 + 8z = 3(11 + 3z) * z

Reorder the terms for easier multiplication:
22 + 8z = 3z(11 + 3z)
22 + 8z = (11 * 3z + 3z * 3z)
22 + 8z = (33z + 9z2)

Solving
22 + 8z = 33z + 9z2

Solving for variable 'z'.

Combine like terms: 8z + -33z = -25z
22 + -25z + -9z2 = 33z + 9z2 + -33z + -9z2

Reorder the terms:
22 + -25z + -9z2 = 33z + -33z + 9z2 + -9z2

Combine like terms: 33z + -33z = 0
22 + -25z + -9z2 = 0 + 9z2 + -9z2
22 + -25z + -9z2 = 9z2 + -9z2

Combine like terms: 9z2 + -9z2 = 0
22 + -25z + -9z2 = 0

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
-2.444444444 + 2.777777778z + z2 = 0

Move the constant term to the right:

Add '2.444444444' to each side of the equation.
-2.444444444 + 2.777777778z + 2.444444444 + z2 = 0 + 2.444444444

Reorder the terms:
-2.444444444 + 2.444444444 + 2.777777778z + z2 = 0 + 2.444444444

Combine like terms: -2.444444444 + 2.444444444 = 0.000000000
0.000000000 + 2.777777778z + z2 = 0 + 2.444444444
2.777777778z + z2 = 0 + 2.444444444

Combine like terms: 0 + 2.444444444 = 2.444444444
2.777777778z + z2 = 2.444444444

The z term is 2.777777778z.  Take half its coefficient (1.388888889).
Square it (1.929012346) and add it to both sides.

Add '1.929012346' to each side of the equation.
2.777777778z + 1.929012346 + z2 = 2.444444444 + 1.929012346

Reorder the terms:
1.929012346 + 2.777777778z + z2 = 2.444444444 + 1.929012346

Combine like terms: 2.444444444 + 1.929012346 = 4.37345679
1.929012346 + 2.777777778z + z2 = 4.37345679

Factor a perfect square on the left side:
(z + 1.388888889)(z + 1.388888889) = 4.37345679

Calculate the square root of the right side: 2.091281136

Break this problem into two subproblems by setting 
(z + 1.388888889) equal to 2.091281136 and -2.091281136.

Subproblem 1

z + 1.388888889 = 2.091281136 Simplifying z + 1.388888889 = 2.091281136 Reorder the terms: 1.388888889 + z = 2.091281136 Solving 1.388888889 + z = 2.091281136 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.388888889' to each side of the equation. 1.388888889 + -1.388888889 + z = 2.091281136 + -1.388888889 Combine like terms: 1.388888889 + -1.388888889 = 0.000000000 0.000000000 + z = 2.091281136 + -1.388888889 z = 2.091281136 + -1.388888889 Combine like terms: 2.091281136 + -1.388888889 = 0.702392247 z = 0.702392247 Simplifying z = 0.702392247

Subproblem 2

z + 1.388888889 = -2.091281136 Simplifying z + 1.388888889 = -2.091281136 Reorder the terms: 1.388888889 + z = -2.091281136 Solving 1.388888889 + z = -2.091281136 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.388888889' to each side of the equation. 1.388888889 + -1.388888889 + z = -2.091281136 + -1.388888889 Combine like terms: 1.388888889 + -1.388888889 = 0.000000000 0.000000000 + z = -2.091281136 + -1.388888889 z = -2.091281136 + -1.388888889 Combine like terms: -2.091281136 + -1.388888889 = -3.480170025 z = -3.480170025 Simplifying z = -3.480170025

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.702392247, -3.480170025}

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